Method of estimating chemical reactivity of nanoparticles

ABSTRACT

The catalytic efficiency of supported catalysts containing metal nanoparticles is strongly related to the chemical softness at the surfaces of such nanoparticles. The chemical softness of a nanoparticle is obtained using results from Density Functional Theory modeling, an extended version of Embedded Atom Method modeling, and continuum modeling based on size and shape of the nanoparticle. A metal nanoparticle of a certain size and shape is first modeled using the extended EAM and EAM parameters that have been validated with results from DFT modeling, to obtain atomic energy densities at each atom location. The chemical softness value at each atom location is then calculated from the atomic energy densities and various parameters that are derived based on results from DFT modeling. The surface chemical softness value is derived from the local chemical softness values based on the geometry and atomistic structure of the metal nanoparticle.

RELATED APPLICATION

This application claims the benefit of Provisional Patent ApplicationNo. 60/629,698, filed Nov. 19, 2004, entitled “Method of PredictingChemical Reactivity of Nanoparticles.”

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention relates to supported catalysts, more specificallyto supported catalysts containing nanometer sized platinum particlesdispersed throughout the catalyst support material.

2. Description of the Related Art

Many industrial products such as fuels, lubricants, polymers, fibers,drugs, and other chemicals would not be manufacturable without the useof catalysts. Catalysts are also essential for the reduction ofpollutants, particularly air pollutants created during the production ofenergy and by automobiles. The majority of industrial catalysts arecomposed of a high surface area support material upon which chemicallyactive metal nanoparticles (i.e., nanometer sized metal particles) aredispersed. The support materials are generally inert, ceramic typematerials having surface areas on the order of hundreds of squaremeters/gram. This high specific surface area usually requires a complexinternal pore system. The metal nanoparticles are deposited on thesupport and dispersed throughout this internal pore system, and aregenerally between 1 and 100 nanometers in size.

The effect of nanoparticle size on catalytic activity is recognized inthe current art. An article by Alexis T. Bell entitled, “The Impact ofNanoscience on Heterogeneous Catalysis,” Vol. 299, Science, Mar. 14,2003, teaches that the reactivity and selectivity of catalystnanoparticles are strongly dependent on their size. FIG. 1 is a diagramfrom this article which shows that the catalytic activity (inparticular, CO oxidation) of Au particles is sensitive to their size andthat only particles in the range of 2 to 3 nm are active.

SUMMARY OF THE INVENTION

In addition to size, chemical softness of nanoparticles, in particular,chemical softness at the surfaces of nanoparticles, affects thecatalytic efficiency of nanoparticles. The present invention provides amethod for modeling the chemical softness of nanoparticles, so that thecatalytic efficiency of such nanoparticles can be quantified in asystematic manner.

The supported catalysts containing platinum nanoparticles having averagesurface softness values (expressed in scaled units ranging from 0 to 1)between 0.07198 and 0.09247 exhibit high catalytic efficiency. Thecatalytic efficiency of such platinum nanoparticles for CO oxidation,expressed as the turn-over frequency (TOF), expressed as per second/perexposed (surface) atom, was observed to be on or above 0.03062 s⁻¹. Bycontrast, the TOF of two prior art nanoparticle samples was observed tobe 0.02946 s⁻¹ and 0.02982 s⁻¹, respectively. The supported catalystscontaining platinum nanoparticles with tighter average surface softnessranges exhibit even higher catalytic efficiencies. The TOF for COoxidation of platinum nanoparticles having average surface softnessvalues (expressed in scaled units ranging from 0 to 1) between 0.08031and 0.08679 was observed to be on or above 0.06554 s⁻¹.

The chemical softness of a nanoparticle is obtained using results fromDensity Functional Theory (DFT) modeling, an extended version ofEmbedded Atom Method (EAM) modeling, and continuum modeling based onsize and shape of the nanoparticle. In the embodiment of the inventiondescribed herein, a platinum nanoparticle of a certain size and shape isfirst modeled using the extended EAM (XEAM) and EAM parameters that havebeen validated with results from DFT modeling, to obtain atomic energydensities at each atom location. The chemical softness value at eachatom location is then calculated from the atomic energy densities andvarious parameters that are derived based on results from DFT modeling.The surface chemical softness value is derived from the local chemicalsoftness values based on the geometry and atomistic structure of theplatinum nanoparticle.

BRIEF DESCRIPTION OF THE DRAWINGS

So that the manner in which the above recited features of the presentinvention can be understood in detail, a more particular description ofthe invention, briefly summarized above, may be had by reference toembodiments, some of which are illustrated in the appended drawings. Itis to be noted, however, that the appended drawings illustrate onlytypical embodiments of this invention and are therefore not to beconsidered limiting of its scope, for the invention may admit to otherequally effective embodiments.

FIG. 1 is a prior art diagram showing the relationship between catalyticactivity and nanoparticle size.

FIG. 2 is a graph showing the catalytic efficiencies of variousnanoparticles, including those nanoparticles that are within the scopeof the present invention.

FIG. 3 is a flow diagram that illustrates the process steps carried outto calculate an average surface softness of a nanoparticle.

FIG. 4 is a flow diagram that illustrates the process steps carried outto calculate average surface softness of a nanoparticle batch.

FIG. 5 is a graph showing the effect of surface softness on catalyticefficiency.

DETAILED DESCRIPTION

FIG. 2 shows the measured TOF for CO oxidation for several platinumnanoparticle batches. These platinum batches represent platinumparticles that are part of alumina-supported platinum catalysts thatwere synthesized according to the methods disclosed in U.S. patentapplication Ser. No. 10/975,646, filed Oct. 28, 2004, the entirecontents of which are incorporated by reference herein. For comparisonpurposes, the measured TOF for CO oxidation for two prior art samples isalso shown.

The physical characteristics of the platinum batches and their measuredTOF for CO oxidation are summarized in Table 1 below. The example columnidentifies the synthesis example that is disclosed in U.S. patentapplication Ser. No. 10/975,646. The support material for all batches isalumina having a BET surface area of 150 m²/g. Other support materialsthat may be used to support platinum nanoparticles include silica,oxides of vanadium, oxides of titanium, oxides of zirconium, oxides ofiron, cerium oxides, carbon, zeolites, and molecular sieves.

TABLE 1 Pt Batch Example Support <D> (nm) loading (wt %) TOF (s⁻¹) PriorArt 1 N/A alumina 2.59364 0.4 0.02982 Prior Art 2 N/A alumina 1.606380.4 0.02946 Pt-48 1 alumina 1.38462 0.4 0.00745 Pt-59 2 alumina 3.685940.4 0.04884 Pt-63 5 alumina 3.15250 0.4 0.05855 Pt-64 6 alumina 1.994500.4 0.02398 Pt-65 7 alumina 2.15752 0.4 0.05157 Pt-68 8 alumina 3.118280.4 0.04845 Pt-69 9 alumina 2.69794 0.4 0.09849 Pt-72 10 alumina 2.340960.4 0.06554 Pt-74 11 alumina 1.91614 0.4 0.05340 Pt-76 12 alumina2.10924 0.4 0.05721 Pt-77 13 alumina 1.98854 0.4 0.02893

The characterization of the platinum batches was carried out bytransmission electron microscopy (TEM). Alternatively, scanningtransmission electron microscopy (STEM) may be used. Prior to themeasurements, a statistically valid sample (about 10-20%) of theplatinum metal particles were prepared using the technique describedbelow and disclosed in U.S. patent application Ser. No. 11/016,578,filed Dec. 17, 2004, the entire contents of which are incorporated byreference herein.

First, the alumina-supported catalyst containing platinum particles isground or milled into fine powder. Then, the powder is mixed intoethynol, and hydrofluoric acid is added to the solvent. The HF acid iseffective in separating the platinum particles from the alumina support.The desired concentration of HF in the resulting solution is about 20%,although HF concentration in the range of 10-50% will also work. Theresulting solution is then placed in an ultrasonic chamber andultrasonic waves are generated and applied to the solution for about 1hour. After letting the solution sit for 12-24 hours, a sample isextracted from the solution and applied to a molybdenum grid that isused by the TEM device. Before the metal grid is placed in the TEMdevice for imaging, the sample applied to the molybdenum grid is dried.

The average diameter <D> of the platinum batches is estimated as twicethe average harmonic parameter. The harmonic parameter is equal to2×(Area of the nanoparticle observed with TEM)/(Perimeter of thenanoparticle observed with TEM). Dimension measurements made by TEM havean estimated error of about 10%.

The TOF for CO oxidation of the platinum batches was measured in thefollowing manner. The batch is first subjected to a standardizedcalcining process. The standardized calcining process includes: (1)loading the batch into a reactor; (2) purging the reactor with He atroom temperature to remove air in the reactor; (3) heating the batch in1% oxygen (remainder inert gas) at a rate of 3° C./minute from roomtemperature to about 500° C.; (4) purging the reactor for 10 minuteswith pure He at 500° C. to remove oxygen; (5) purging the reactor in 5%hydrogen (remainder inert gas) for 1 hour at 500° C.; and (6) purgingthe reactor in pure He while cooling down the reactor to roomtemperature. Then, without removing the batch from the reactor, the COoxidation is carried out. The CO oxidation process includes: (1) purgingthe reactor with the reaction mixture of 1.4% CO, 5.6% O₂ (balance He)at room temperature; and (2) heating the reactor from room temperatureto 200° C. at about 2° C./minute with the aforementioned CO/O₂ mixture.During this heating step, CO₂ yield is measured as a function of thetemperature.

${TOF} = {\frac{{Total}\mspace{14mu}{CO}_{2}\mspace{14mu}{yield}\mspace{14mu}{at}\mspace{14mu} 125{^\circ}\mspace{14mu}{C.}}{N} \times \frac{N}{N_{s}}}$

In the equation above, N is the total number of atoms and N_(S) is thetotal number of surface atoms. These values are derived using the areameasurements from TEM and r_(M), the metallic radius of the atom.

Table 1 shows that the TOF for platinum batches with similar averagediameters, e.g., Pt-64 vs. Pt-74, may vary quite a bit. This indicatesthat size alone is a poor predictor of catalytic performance of ananoparticle. The present invention takes into account additionalfactors with the goal of more accurately predicting the catalyticperformance of a nanoparticle. One such additional factor is chemicalsoftness, in particular, the average chemical softness at the surfacesof the nanoparticle (also referred to herein as “average surfacesoftness”).

The process for determining the average surface softness is illustratedin the flow diagram of FIG. 3. In step 31, a nanoparticle of a certainelement is selected for modeling. The crystal structure of the selectedelement is taken from available crystallographic tables at the elementalenergy ground state. The element may be any element, including platinum,silver, copper, palladium and any other metallic element that iscommonly used as a catalyst metal. The size corresponds to the size ofthe actual synthesized sample that is being analyzed, as determined fromTEM characterization of a statistically valid sample (e.g., 10-20%)described above, or a theoretical sample that is being modeled. Theshape corresponds to the shape of the selected element at its groundstate. For platinum, this shape is truncated octahedron. Based on theelement and its crystal structure, size and shape, the number of atomsand the geometric locations of the atoms of the nanoparticle can bederived.

In step 32, the atomic energy density at each atom location of thenanoparticle, E_(i), is calculated using an extended version of the EAM.The EAM provides the following formulations for E_(i):

${E_{i}{\sum\limits_{j{({\neq i})}}\;{\Phi( R_{ij} )}}} + {F_{i}( \rho_{i} )}$

The XEAM extends the above formulations for the EAM in the followingmanner:

$E_{i} = {{\sum\limits_{j{({\neq i})}}\;{V( r_{ij} )}} + {( {1 - {0.5*\frac{\rho_{i}^{asym}}{\rho_{i}}}} ){F_{i}( \rho_{i} )}}}$

The function Φ(R_(ij)) is the pair potential function in the EAMformulation and the function F_(i)(ρ_(i)) is the embedding function inthe EAM formulation. The symbol, ρ, represents the (modeled) chargedensity function. The function, ρ_(i), represents the charge densityfunction at atom location i, and the function, ρ_(i) ^(asym), representsthe asymmetric charge density function at atom location i. Theformulations of ρ_(i) and ρ_(i) ^(asym) are set forth below:

${\rho_{i} = {\sum\limits_{j{({\neq i})}}\;{\rho_{j}^{a}\lbrack R_{ij} \rbrack}}};\mspace{14mu}{{{and}\mspace{14mu}\rho_{i}^{asym}} = {{{\sum\limits_{j{({\neq i})}}{{\rho_{j}^{a}\lbrack R_{ij} \rbrack}\frac{R_{ij}}{{R_{ij}}}}}}}}$where ρ_(j) ^(a) is the charge density contribution coming from site jto the atom at site i. A total of five EAM parameters are used in theEAM. They are: χ, α, β, F₁ and r_(a). The use of these parameters inaccordance with EAM and as applied to seven face-centered cubic (fcc)metals (Al, Ag, Au, Cu, Ni, Pd, and Pt) and their binary alloys isdescribed in an article by J. Cai and Y. Ye, “Simple analyticalembedded-atom-potential model including a long-range force for fccmetals and their alloys,” Phys. Rev. B, Vol. 54, p. 8398 (1996), theentire contents of which are incorporated by reference herein.

In the embodiment of the invention described herein, these parametersare derived to reproduce the energy density computed using DFT modelingand have the following values for platinum:χ=4.3 Å⁻¹;α=0.4033 eV;β=5.6379;F₁=0.6815 eV; andr_(a)=2.3839 Å.

In step 33, the chemical softness at each atom location of thenanoparticle, s_(i) is calculated using the following formulation:

$s_{i} = {s_{gs} + {( {s_{at} - s_{gs}} ){\sum\limits_{n = 1}^{5}\;{C_{n}( \frac{E_{i} - E_{gs}}{E_{at} - E_{gs}} )}^{n}}}}$where:the chemical softness, s_(i), is unit-less;s_(gs) is the softness for the bulk atoms (set to 0);s_(at) is the softness of a free atom is derived experimentally or itmay be calculated using DFT; this value is set to 1 and all othersoftness values are scaled with respect to this value;{C₁, C₂, C₃, C₄, C₅} are universal constants that are used to model anyelement including platinum;C₁=0.00031671;C₂=2.03164;C₃=−0.0198892;C₄=−6.60821; andC₅=5.04367.E_(gs) is the atomic energy density for the bulk atoms (this value isdifferent for different metals; for platinum, E_(gs)=−5.7 eV, but forsilver, E_(gs)=−2.8 eV); andE_(at) is the atomic energy density for a free atom (set to 0).

In step 34, the average surface softness is derived from the localchemical softness values based on the geometry and atomistic structureof the nanoparticle. As part of this derivation, the computed localsoftness value for each atom, s_(i), is first distributed around thatvalue using a Gaussian distribution with a spread, σ, to model thesurface imperfections due to temperature. The spread, σ, is zero forideal conditions (e.g., T=0° K). Otherwise, the spread, σ, is equal tok_(B)*T, where k_(B) is the Boltzmann constant and T is temperature in °K at which CO₂ yield is measured to compute TOF. The Gaussiandistributions are then summed to produce the softness profile N(s) thatspecifies the number of atoms (N) corresponding to a softness value, s.The softness profile, N(s), can be expressed in the following equationform:

${N(s)}{\sum\limits_{i = 1}^{Ntotal}\;{\frac{1}{\sigma\sqrt{2\pi}}{\exp( {- \frac{{{s - s_{i}}}^{2}}{2\sigma^{2}}} )}}}$where Ntotal is the total number of atoms in the nanoparticle beingmodeled.

The average surface softness, s_(avg), is derived from the softnessprofile, N(s), using the following equation:

$S_{avg} = \frac{\int{{\mathbb{d}s}\;{f(s)}s\;{N(s)}}}{\int{{\mathbb{d}s}\;{f(s)}\;{N(s)}}}$where f(s) is the filtering function that filters out the softnessvalues associated with the bulk atoms.

The process for determining the average surface softness of ananoparticle batch is illustrated in the flow diagram of FIG. 4. In step41, a nanoparticle batch of a certain element is selected for modeling.The element may be any element, including platinum, silver, copper,palladium and any other metallic element commonly used as a catalystmetal. The size distribution of the nanoparticles in the batch isdetermined from TEM characterization of a statistically valid sample(about 10-20%) as described above. The shape of the nanoparticles in thebatch is assumed to be the lowest energy state shape. When modelingplatinum nanoparticles, the shape is assumed to be truncated octahedron,which has the lowest energy state. The number of atoms and the geometriclocations of the atoms of any one nanoparticle in the sample can bederived based on that nanoparticle's element, crystal structure, size,and shape.

In step 42, the process steps 32-33 of FIG. 3 are carried out for eachof the nanoparticles. Then, in step 43, the softness profile, N(s), iscalculated from the local softness values obtained in step 42. In step44, the average surface softness of the nanoparticle batch is calculatedbased on the softness profile.

The TOF curve plotted against surface softness exhibits a volcano curvein the shape of a Lorentzian function. Table 2 shows different platinumnanoparticle batches and the resulting average surface softness values(expressed in scaled units ranging from 0 to 1) that fall on the volcanocurve in the shape of a Lorentzian function. The “TOF vs. softness”volcano curve is shown in FIG. 5.

TABLE 2 Batch Example Support <D> (nm) TOF (s⁻¹) <surface softness>Pt-48 1 alumina 1.38462 0.00745 0.10110 Pt-59 2 alumina 3.68594 0.048840.07516 Pt-63 5 alumina 3.15250 0.05855 0.07198 Pt-64 6 alumina 1.994500.02398 0.09274 Pt-65 7 alumina 2.15752 0.05157 0.07818 Pt-68 8 alumina3.11828 0.04845 0.07674 P1-69 9 alumina 2.69794 0.09849 0.08031 Pt-72 10alumina 2.34096 0.06554 0.08679 Pt-74 11 alumina 1.91614 0.05340 0.09247Pt-76 12 alumina 2.10924 0.05721 0.08973 Pt-77 13 alumina 1.988540.02893 0.09530

While particular embodiments according to the invention have beenillustrated and described above, those skilled in the art understandthat the invention can take a variety of forms and embodiments withinthe scope of the appended claims.

1. A method of estimating chemical reactivity of a particle, comprisingthe steps of: determining size and shape of the particle; approximatingatomic energy densities at various locations of atoms in the particle;calculating local chemical softness values at said various locations ofatoms based on the atomic energy densities; calculating an averagesurface chemical softness of the particle based on the local chemicalsoftness values; and estimating the chemical reactivity of the particlebased on the average surface chemical softness value of the particle. 2.The method according to claim 1, wherein the shape of the particle isapproximated as the shape of the particle at its ground state.
 3. Themethod according to claim 2, wherein the particle comprises platinum andthe shape of the particle is truncated octahedron.
 4. The methodaccording to claim 1, further comprising the step of determining thenumber of atoms and geometric locations of said atoms based on the sizeand shape of the particle.
 5. The method according to claim 1, whereinthe particle comprises platinum and the local chemical softness valuesare further based on the chemical softness of a free platinum atom andthe atomic energy density of bulk platinum atoms.
 6. The methodaccording to claim 1, wherein the atomic energy densities areapproximated using an embedded atom method.
 7. The method according toclaim 6, wherein parameters used in the embedded atom method are derivedto reproduce the energy densities computed using DFT modeling.
 8. Themethod according to claim 1, wherein the chemical reactivity datacomprises catalytic activity data, and the catalytic activity data ofthe particle is estimated based on catalytic activity data of otherparticles with similar average surface chemical softness values.
 9. Amethod of estimating chemical reactivity of a metal particle batch,comprising the steps of: extracting a statistically valid sample of themetal particle batch; determining size and shape of each of theparticles in the sample; for each particle in the sample, calculatinglocal softness values at various atom locations; calculating an averagesurface softness of the metal particle batch based on the local softnessvalues of the particles in the sample; and estimating the chemicalreactivity of the metal particle batch based on chemical reactivity dataof other metal particle batches with similar average surface softnessvalue.
 10. The method according to claim 9, wherein each step ofcalculating the local surface softness values comprises the steps of:approximating atomic energy densities at various locations of atoms inthe particle; and calculating local softness values at said variouslocations of atoms based on the atomic energy densities.
 11. The methodaccording to claim 10, wherein the metal particle comprises platinum andthe local chemical softness values are further based on the chemicalsoftness of a free platinum atom and the atomic energy density of bulkplatinum atoms.
 12. The method according to claim 10, wherein the atomicenergy densities are approximated using an embedded atom method.
 13. Themethod according to claim 9, wherein the shape is truncated octahedron.14. A method of estimating catalytic efficiency of a metal particle,comprising the steps of: calculating an average surface softness of themetal particle based on the structure of the metal particle; measuringthe catalytic efficiency of various samples of the metal particle havingdifferent structure; calculating average surface softness of each of thevarious samples; and estimating the catalytic efficiency of the metalparticle based on the average surface softness of the metal particle andthe relationship of the catalytic efficiency to the average surfacesoftness of the various samples.
 15. The method according to claim 14,wherein the measured catalytic efficiency comprises turn-over frequency.16. The method according to claim 15, wherein the step of measuringcomprises the steps of subjecting each sample to CO oxidation andmeasuring the CO₂ yield; and deriving TOF based on the total CO₂ yielddivided by the number of surface atoms on the sample.
 17. The methodaccording to claim 16, wherein the step of measuring further comprisesthe step of determining the number of total atoms in the sample and theTOF is derived based on the number of total atoms in the sample and TEMcharacterizations of area and perimeter of the samples.
 18. The methodaccording to claim 14, wherein each step of calculating the averagesurface softness comprises the steps of: approximating atomic energydensities at various locations of atoms in the metal particle;calculating local chemical softness values at said various locations ofatoms based on the atomic energy densities; and calculating the averagesurface chemical softness of the metal particle based on the localchemical softness values.
 19. The method according to claim 18, whereinthe metal particle comprises platinum and the local chemical softnessvalues are further based on the chemical softness of a free platinumatom and the atomic energy density of bulk platinum atoms.
 20. Themethod according to claim 18, wherein the atomic energy densities areapproximated using an embedded atom method.